Dear Georges Elencwajg, yes, and even if the decomposition is actually unique one can cause confusion. Some people like to tease kids by pronouncing the following wrong and asking them whether they know what it means: Blumento|pferde (suggesting Blumento-horses which do not exist) instead of Blumentopf|erde (flower-pot soil); same for Palat|schinken (suggesting it is Palat-ham which does not exist) which is actually just one word Palatschinken (some form of crepe, derived from Czech, Hungarian, or Romanian).
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quidApr 19 '11 at 16:32

I did not know that, it's very cool! Well, I suspected that the F stood for "fermé", but then you tend to automatically assume that G was used because it's the next letter available... So far, this has to be my favorite answer.
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Thierry ZellApr 19 '11 at 15:30

Ansatz. Although I suppose it is used more in physics than in mathematics. I don't know why the translation is not used often, but I guess it has to do something with the fact that in the beginning of the 20th century German was used much more than English in the scientific literature, I believe.

Well maybe not a literal translation, but in most cases "educated guess" can be used as well.
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Pieter NaaijkensApr 19 '11 at 11:16

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@Pieter: Maybe that this is really the closest english expression, but personally I would never use it, because an "Ansatz" has a very different feeling than a "Schuss ins Blaue" / "educated guess". If you have an Ansatz, you have an idea of what is going on or should be going on. Maybe you have physical reasons to believe that the solution of your equation should have a particular nice form or something like that. An educated guess on the other hand is ... well, guessing. And that is a very different kind of approach I think.
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Johannes HahnApr 19 '11 at 15:13

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@Johannes Hahn, while I tend to agree that 'Ansatz' is not really an 'educated guess' I would say 'Schuss ins Blaue' is neither; as this I believe is more a 'shot in the dark' so a 'wild guess'.
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quidApr 19 '11 at 15:26

This is an answer to the part of the question about why these terms are not translated into English. The reason is that words such as "nullstellensatz", "Schadenfreude" and so on that you mistakenly think are German are in fact perfectly good English words and so do not need translation. (Look up Schadenfreude in the Oxford English Dictionary if you do not believe it is an English word, though they have not yet caught up with nullstellensatz.) The point is that unlike languages such as French and German that try to remain pure, English has been happily looting terms from other languages for centuries, and the only difference between "nullstellensatz" and "house" is that "house" was stolen so long ago that we have forgotten about it.

English has been happily looting... For example the word loot (लूट) from us Indians.
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Chandan Singh DalawatApr 19 '11 at 14:26

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Whether this is done more 'happily' in English, I don't know. But, it is certainly also done in German, with certain regional variations in the extent (three of the italic words might not be common everywhere, but the others are universal several even without common alternative): "Von der Trafik aus, flanierte ich ueber das Trottoir, einen salutierendenOffizier mit Pistole und einen Portiernonchalantpassierend, ins Souterrain." And, in France you might well wish your collegues after a 'planning' on friday afternooon 'bonne week-end'.
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quidApr 19 '11 at 15:07

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@Chandan: one can imagine Indians using that word a lot in talking with/about the Englsh, initially :)
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Mariano Suárez-Alvarez♦Apr 19 '11 at 17:04

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Sorry, but "house" most certainly did not come into English from German. The word hus occurs in every one of the oldest Germanic languages. By standard sound-changes, the long "u" shifted to "ou" in English, just as "ut" became "out", "mus" became "mouse", etc.
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LubinApr 20 '11 at 15:03

The word idele often is written with an accent on the first e, hinting at its French origin (Chevalley introduced ideal elements, and Hasse suggested the word idel; see p. 91 in Emil Artin und Helmut Hasse: die Korrespondenz 1923 - 1934, available online).
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Franz LemmermeyerApr 19 '11 at 14:50

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It is true that the word idèle was introduced by Chevalley but it was based on the German contraction "id. ele." for ideales Element. If it has been based on a French contraction, it would have been éléïde.
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Chandan Singh DalawatApr 20 '11 at 2:33

The term "Umkehrabbildung" (Abbildung being the german word for map) usually refers to and is translated as "inverse map". Do you know why are pushforwards are named like this? I mean $f_\ast$ is usually not invers to $f$ or even to $f^\ast$...
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Johannes HahnApr 19 '11 at 15:05

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@Johannes: Usually you use the term Umkehr map when a map going in the other direction is much easier to define, hence the “reversal”.
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Dmitri PavlovApr 19 '11 at 18:56

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Strangely enough, many topologists in Germany tend to call umkehr maps "pushforward". Maybe because otherwise they could be confused with inverse maps; maybe "pushforward" just sounds more faashionable.
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Johannes EbertApr 20 '11 at 8:23

You're probably right, but the truth should never interfere with a good story. What I heard is that the (contravariant functor) K comes from grothendiecK. There is also a covariant G.
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Donu ArapuraApr 19 '11 at 11:19

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That the K of K-theory comes from Klasse is explained by Grothendieck himself in the first issue of the journal K-theory. He first wanted to use C, from the French word "classe", but being an analyst he feared that it would cause a confusion with $C(X)$, the continuous functions on $X$. So he decided to use the initial of the translation of "classe" in his native German, Klasse.
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Georges ElencwajgApr 19 '11 at 16:28

In German I never heard this called anything else than 'Kleinsche Flasche' (I thus somewhat doubt there was the development you sketch).
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quidApr 19 '11 at 11:33

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It was then retranslated to German as Kleinsche Flasche which I omitted above because the question asks about usage in English. So unless you are 120 years old, you have no chance to have heard Kleinsche Fläche.
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user11235Apr 19 '11 at 11:35

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Here the commenting unknown again: sorry for my initial doubts. What you write certainly agrees with the German Wikipedia entry, though that entry is a bit vague (as it is more or less reportes a rumour). Perhaps I will try to find some more information. If this turns out to be true it is a quite fun development. So, +1.
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quidApr 19 '11 at 11:43

Very interesting! I wonder which came first---the term "Klein bottle," or the standard bottle-shaped immersion of the Klein bottle? To me, the picture of the Klein surface in the book Tara Brough linked is quite striking---I don't think I've ever seen the Klein surface drawn that way, even though it's a perfectly natural way of doing it.
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VectornautApr 19 '11 at 21:41

Plastikstufe = a certain higher dimensional analogue of an overtwisted disk in contact geometry. This is not a real German word. It is a compound of the German words for "plastic" and "step", but this does not have any obvious relevance to its mathematical meaning. There is a funny story about where this word came from which however is not appropriate for this forum.

@Yemon: apparently yes. From wikipedia: "The term finds its formal definition in combinatorial game theory, and it describes a situation where one player is put at a disadvantage because he has to make a move when he would prefer to pass and make no move."
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Thierry ZellApr 20 '11 at 1:50

I've seen schlicht-function for functions $f(x)=x +a x^2 + b x^3 + ...$ for powerseries without constant term and $f'(0)=1 $. But I do not really know, whether this is really the german word schlicht (=simple) or only some coincidence.

It certainly is the german word schlicht (maybe innocuous would be a more adequate translation than simple). Conformal mapping was dominated by the German school (Koebe, Bieberbach) before the Finnish school took over. However, univalent seems to be the preferred term nowadays.
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Theo BuehlerApr 19 '11 at 9:20

Gentzen's Hauptsatz (cut elimination theorem) : This is a fundamental result in structural proof theory, and is at the heart of Gentzen's consistency proof of elementary number theory. It is very funny that the word literally means "main theorem," with no reference to the subject domain, yet it is standard in logic in English to use just the word "Hauptsatz" to refer to this (family of) theorem(s) in proof theory.

Stufe (=level) of a non-real field (wikipedia.de). It is the least number of squares $a_i^2$ such that $\sum_i a_i^2 = -1$, $\infty$ if no such sum exists.

In this paper, the level of a subgroup of $SL_2(\mathcal{O})$ is defined ($\mathcal{O}$ a number field), as the generalisation of the stufe of a field, so the term has been translated, but only in a shift of context.

Viergeflechte, the original German name for 2-bridge knots, still occasionally used in an English context. In his Mathematical Review of Schubert's 1956 paper "Knoten mit 2 Bruecken" Fox explicitly notes that "Viergeflecht" is untranslatable.